Initial Motion of Boulders in Bedrock Channels

The behaviour of a viscous vortex ring is examined by a matched asymptotic analysis up to three orders. This study aims at investigating how much the location of maximum vorticity deviates from the centroid of the vortex ring, defined by P. G. Saffman (1970). All the results are presented in dimensionless form, as indicated in the following context. Let Γ be the initial circulation of the vortex ring, and R denote the ring radius normalized by its initial radius R i . For the asymptotic analysis, a small parameter ∊ = ( t / Re ) ½ is introduced, where t denotes time normalized by R 2 i / Γ , and Re = Γ/v is the Reynolds number defined with Γ and the kinematic viscosity v . Our analysis shows that the trajectory of maximum vorticity moves with the velocity (normalized by Γ/R i ) U m = – 1/4π R {ln 4 R /∊ + H m } + O (∊ ln ∊), where H m = H m ( Re, t ) depends on the Reynolds number Re and may change slightly with time t for the initial motion. For the centroid of the vortex ring, we obtain the velocity U c by merely replacing H m by H c , which is a constant –0.558 for all values of the Reynolds number. Only in the limit of Re → ∞, the values of H m and H c are found to coincide with each other, while the deviation of H m from the constant H c is getting significant with decreasing the Reynolds number. Also of interest is that the radial motion is shown to exist for the trajectory of maximum vorticity at finite Reynolds numbers. Furthermore, the present analysis clarifies the earlier discrepancy between Saffman’s result and that obtained by C. Tung and L. Ting (1967).

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Discussion of “Initial Motion Problem in Porous Media”

Journal of the Hydraulics Division ◽

10.1061/jyceaj.0001368 ◽

1965 ◽

Vol 91 (6) ◽

pp. 221-223

Author(s):

Gedeon Dagan

Keyword(s):

Porous Media ◽

Motion Problem ◽

Initial Motion

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2. On Vortex Motion

Proceedings of the Royal Society of Edinburgh ◽

10.1017/s0370164600042632 ◽

1872 ◽

Vol 7 ◽

pp. 576-577

Author(s):

William Thomson

Keyword(s):

Royal Society ◽

Vortex Motion ◽

Bounding Surface ◽

Initial Shape ◽

Liquid Mass ◽

Dissipation Of Energy ◽

Surrounding Liquid ◽

Initial Motion

AbstractThis paper is a sequel to several communications which have already appeared in the Proceedings and Transactions of the Royal Society of Edinburgh. It commences with an investigation of the circumstances under which a portion of an incompressible frictionless liquid, supposed to extend through all space, or through space wholly or partially bounded by a rigid solid, can be projected so as to continue to move through the surrounding liquid without change of shape; and goes on to investigate vibrations executed by a portion of liquid so projected, and slightly disturbed from the condition that gives uniformity. The greatest and least quantities of energy which a finite liquid mass of any given initial shape and any given initial motion can possess, after any variations of its bounding surface ending in the initial shape, are next investigated; and the theory of the dissipation of energy in a finite or infinite frictionless liquid is deduced.

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Review: State of the art study of the influence of bed roughness and alluvial cover on bedrock channels and comparisons of existing models with laboratory scale experiments

10.5194/esurf-2019-78-rc2 ◽

2020 ◽

Author(s):

Rebecca Hodge

Keyword(s):

State Of The Art ◽

Laboratory Scale ◽

Bed Roughness ◽

Review State ◽

Bedrock Channels

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The initial motion of a gas bubble formed in an inviscid liquid Part 1. The two-dimensional bubble

Journal of Fluid Mechanics ◽

10.1017/s0022112062000300 ◽

1962 ◽

Vol 12 (3) ◽

pp. 408-416 ◽

Cited By ~ 72

Author(s):

J. K. Walters ◽

J. F. Davidson

Keyword(s):

Constant Pressure ◽

Bubble Radius ◽

Steady Motion ◽

Two Dimensional ◽

Spherical Cap ◽

Liquid Part ◽

Irrotational Motion ◽

Acceleration Of Gravity ◽

Small Bubbles ◽

Initial Motion

The paper deals with the initial motion of a two-dimensional bubble starting from rest in the form of a cylinder with its axis horizontal. The theory is based on the assumptions of irrotational motion in the liquid round the bubble, constant pressure within the bubble, and small displacements from the cylindrical form. This theory predicts that the bubble should rise with the acceleration of gravity, over a distance of at least the initial bubble radius, and that a tongue of liquid should be projected up from the base of the bubble into its interior. These predictions are confirmed by experiments which also show how the vorticity necessary for steady motion in the spherical-cap form is generated by the detachment of two small bubbles from the back of the main bubble.

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A probabilistic framework for the cover effect in bedrock erosion

10.5194/esurf-2016-59 ◽

2016 ◽

Author(s):

Jens M. Turowski ◽

Rebecca Hodge

Keyword(s):

Time Scales ◽

Channel Morphology ◽

Scale Model ◽

Probabilistic Framework ◽

Channel Dynamics ◽

Base Level ◽

Bedrock Channel ◽

Bedrock Erosion ◽

Long Time ◽

Bedrock Channels

Abstract. The cover effect in fluvial bedrock erosion is a major control on bedrock channel morphology and long-term channel dynamics. Here, we suggest a probabilistic framework for the description of the cover effect that can be applied to field, laboratory and modelling data and thus allows the comparison of results from different sources. The framework describes the formation of sediment cover as a function of the probability of sediment being deposited on already alleviated areas of the bed. We define benchmark cases and suggest physical interpretations of deviations from these benchmarks. Furthermore, we develop a reach-scale model for sediment transfer in a bedrock channel and use it to clarify the relations between the sediment mass residing on the bed, the exposed bedrock fraction and the transport stage. We derive system time scales and investigate cover response to cyclic perturbations. The model predicts that bedrock channels achieve grade in steady state by adjusting bed cover. Thus, bedrock channels have at least two characteristic time scales of response. Over short time scales, the degree of bed cover is adjusted such that they can just transport the supplied sediment load, while over long time scales, channel morphology evolves such that the bedrock incision rate matches the tectonic uplift or base level lowering rate.

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